We consider a fluid system in which during off-times the buffer content increases as a piecewise linear process according to some general semi-Markov process, and during on-times, it decreases with a state-dependent rate (or remains at zero). The lengths of off-times are exponentially distributed. We show that such a system has a stationary distribution which satisfies a decomposition property where one component in the decomposition is associated with some dam process and the other with a clearing process. For the cases of constant and linear decrease rates, the steady-state Laplace–Stieltjes transform and moments of the buffer content are computed explicitly.